ABSTRACT
Let F $ \mathcal F $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math4_1.tif"/> be a family of convex sets in R $ \mathbb R $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math4_2.tif"/> . A geometric transversal is an affine subspace that intersects every member of F $ \mathcal F $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math4_3.tif"/> . More specifically, for a given integer 0 ≤ k < d $ 0\le k <d $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math4_4.tif"/> , a k-dimensional affine subspace that intersects every member of F $ \mathcal F $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math4_5.tif"/> is called a k -transversal to F $ \mathcal F $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math4_6.tif"/> . Typically, we are interested in necessary and sufficient conditions that guarantee the existence of a k-transversal to a family of convex sets in R $ \mathbb R $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math4_7.tif"/> , and furthermore, to describe the space of all k-transversals to the given family. Not much is known for general k and d, and results deal mostly with the cases k = 0 $ k=0 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math4_8.tif"/> , 1, or d - 1 $ d-1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math4_9.tif"/> .
